In modern physics, the quantum wave distribution function necessarily uses complex numbers to represent itself. If physics defines the physical reality, then what we are saying by the statement above is that the reality is made up of immeasurable and undefinable complex numbers. In other words, the probability wave function or reality can not be understood natively as represented.

To illustrate, let us consider a statement: there are i mangoes (where i is a complex number). The i mangoes statement can not be understood natively. However if I say i mangoes were distributed to i people then it makes some sense as i multiplied by i gives -1. But neither the i mangoes nor the i people makes any sense.

In Engineering, complex numbers are nothing but a tool to calculate efficiently.
The equations in engineering, which use complex numbers, can be rewritten as real numbers, but in Physics complex numbers are made intrinsic part of reality, thus making reality impossible to understand.

Neither can you ever have pi mangoes. Irrational numbers are also just a mathematical construct. We use these numbers to describe reality, even if they don't necessarily manifest.
– M.HerzkampMar 15 '18 at 10:52

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Putting their name aside, what makes you think that real numbers are an integral part of the physical reality?
– sureMar 15 '18 at 13:09

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1. physics doesn't 'define' reality: it attempts to 'describe' it. 2. complex numbers are not 'undefinable', they have a specific definition like any other proper mathematical construct.
– JimmyJamesMar 15 '18 at 13:39

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As an aside, I challenge the implied presupposition that natural numbers of mangoes are inherently physical. Natural counting is certainly intuitive, but it presupposes that we can clearly and unambiguously identify mangoes, separating them into individual objects to count. I suggest that this is non-trivial and only appears obvious by virtue of the way our cognition and perception function.
– Dan BryantMar 15 '18 at 14:34

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Complex numbers are nothing but pairs of real numbers. Real numbers are nothing but sets of rational numbers. Rational numbers are nothing but pairs of integers. Integers are nothing but pairs of natural numbers. Natural numbers are nothing but sets of sets. So really, you are asking why nature can be represented using nothing but set theory.
– chepnerMar 15 '18 at 17:26

12 Answers
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Complex numbers are not, as you suggest, "...an integral part of physical reality". Neither, as you say, does the "quantum wave distribution function necessarily uses complex numbers". Not necessarily. Quantum mechanics can be mathematically formulated using the real numbers, the complex numbers, or the quaternions. See, e.g., https://arxiv.org/abs/1101.5690 for a mathematical discussion (in particular, see Section 2.4 discussing Soler's theorem, briefly summarized by, e.g., https://en.wikipedia.org/wiki/Sol%C3%A8r%27s_theorem wikipedia).

Although, as per that arxiv cite, complex numbers seem to be most convenient, they're not fundamentally necessary, and have no particular fundamental physical significance. The one-sentence reason why the "quantum wavefunction" (the example you elaborate) conveniently uses complex numbers is because the wavefunction is characterized not only by an amplitude, but also by a phase. And complex numbers conveniently encode the mathematical amplitude,phase relationship. But if you want to represent it somewhat less conveniently, no problem.

In fact, as per my preceding complex number reply, electromagnetic waves are typically also described using complex numbers. Indeed, like I suggested, pretty much any phenomenon described by an amplitude-plus-phase wave will have a convenient complex number representation.

This is no more magical, no more fundamental, than using numbers to count, say, apples (or mangoes as illustrated by @Geoffrey). Numbers are convenient for apple-counting because when you have two apples, and then somebody gives you two more apples, you find that you have ... four apples. And the 2+2=4 algebraic property of numbers conveniently represents the observable behavior of apple accumulation. Nothing more. And neither nothing more about complex numbers in situations where they're convenient.

Edit: since there seems to be more interest in this topic than I'd have thought (657 views as I'm writing), let me elaborate just a bit on my emphasized "any phenomenon described by an amplitude-plus-phase wave will have a convenient complex number representation" remark above. Actually, let me just point you to another stackexchange answer where the idea's much better illustrated than anything I could do...https://electronics.stackexchange.com/questions/128989/
...It's the very pretty animated pictures that illustrate the ideas. It's that two-component (real and imaginary components) "phasor" at the bottom that's used to generate the waveform at top. And there you go -- as you can see from the animations, those two-component complex number phasors capture the entire waveform behavior at one fell swoop. Very convenient. But not physical. The physical stuff is the waveform at top. The complex-number phasor at bottom is just a convenient mathematical way to quantitatively get it. You'll note that the author first discusses "phase" (in the same sense I used it above) and then introduces "phasor" derived from it. If further interested, wikipedia has a longer phase/phasor discussion (and another pretty animated diagram) https://en.wikipedia.org/wiki/Phasor

Right. Complex numbers are just 2D numbers, and in many circumstances 2D numbers are useful. Just as in many circumstances values of even higher dimension are also useful.
– curiousdanniiMar 15 '18 at 6:27

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@curiousdannii 'Complex numbers are just 2D numbers'. No, elements of R^2 are 'just 2D numbers'. Complex numbers are 2D numbers with a convenient multiplication method defined on them. The latter property is essential for its convenient usage here, just as the multiplication is essential for the applicability of quaternions.
– Discrete lizardMar 15 '18 at 14:05

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@Discretelizard I think curiousdannii was trying to say complex numbers are similar to vectors.
– jkdMar 15 '18 at 23:51

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I don't feel like fleshing this out into an answer, but the answers that say the complex numbers aren't an integral part of reality seem to be missing the point. "The complex numbers" are just notation, but the fact that there is field structure that appears when modeling physical phenomena involving phase/amplitude and similar concepts, rather than just an unstructured two-dimensional vector space over the reals, does seem to say something about inherent mathematical reality.
– R..Mar 16 '18 at 16:59

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@R.. I pretty much entirely agree with you. And this whole broadly-defined area has been widely discussed, perhaps ever since Wigner's remark about "the unreasonable effectiveness of mathematics" (google that phrase for lots more links), e.g., dartmouth.edu/~matc/MathDrama/reading/Wigner.html I particularly (but not exclusively) like physicist/philosopher's Max Jammer's analysis of a theory as a "partially interpreted formal system" in chapter 1 of books.google.com/… But there's way too much to say for a comment.
– John ForkoshMar 17 '18 at 1:06

The short answer: Your premise is not correct. Quantum Mechanics is not necessarily complex-valued. Here is a primer from Physics.SE if you are solid on the math.

An explanation that is light on math: Complex numbers represent a particular collection of symmetries that behave in a particular way. They happen to be closely related to Real numbers because real numbers encode information about size and directionality in one dimension while Complex numbers do this in two dimensions. The number "i" is actually a sort of mathematical shorthand for "rotate 90 degrees counterclockwise." This has the upshot that 2-D vectors and traditional 2-D vector algebra can be simply and cleanly represented by complex numbers and complex algebra.

The important thing about quantum theory is that states are no longer coupled to observables as they are in classical physics. Now, the state a particle is in can mix and combine with other states freely, and the observables have no value until measured. Complex numbers (since they add extra "room") encode this mixing potential in a convenient way.

I would recommend that you think about mathematics as being the "science of thought." Every mathematical idea was invented by someone to systematically describe something. This means that when a mathematical idea doesn't generalize to a "common sense" situation (like "i" mangoes) then that means you have removed it from its intended realm of application. Natural numbers are good for counting mangoes because they act like mangoes; complex numbers are good for describing wave functions because (in a way) they behave like wave functions. Try not to put the cart before the horse.

@DheerajVerma No, it doesn't (although I can see why you might read it that way); It's only stating the same thing the answers here say: There's nothing fundamentally "physical" about complex numbers (or real numbers, or any numbers). Complex numbers are just the most convenient way (we know of) to represent these phenomena.
– CubicMar 15 '18 at 11:01

This answer fails to account, for the fact that consequences of axioms needed for the math to work often have physical consequences. Consider Dirac's prediction of antiparticles. That linked thread has the key explanatory fact: "the difference is that the amplitude of a sound wave is an observable, while only the amplitude of the modulus squared is an observable in quantum mechanics. I can see the phase of a water wave, but I can only see the phase of an electron wave through interference effects" It is a probability wave, waves interfere, but observables 'collapse' the probability distrbution
– CriglCraglMar 15 '18 at 14:26

Yeah, instead of C one could just use R^2 with suitable operations.
– TrilarionMar 16 '18 at 16:21

@CriglCragl That's just selection bias. There's plenty of "predictions" based on the math of a model that don't come through, and the fact that some of those predictions actually do represent some reality is relatively rare - but it's far more interesting to talk about the few successes than the far greater number of nonsense. Sure, the failure is a sign that the model is incomplete, but of course the model is incomplete. It's still a very useful tool for probing the frontiers of a model, mind you, but don't get too carried away - it's only ever as good as the model.
– LuaanMar 19 '18 at 12:46

You are right that quantum mechanics is based on the psi-function which is a complex function. The squared modulus of this function, a real number between zero and one, is the probability distribution of particles. Only the latter can be measured. But the mathematical formalism of the Schroedinger equation is based on the complex psi-function. The real probability function is not sufficient. To understand nature we have to learn which means are suitable to apply. Nature does not follow our predilections.

Complex numbers, in particular imaginary numbers, are definable and understandable. Concerning the definition: A complex number has a real part and an imaginary part: z = x+iy. It is possible to add, subtract, multiply and divide complex numbers similar to real numbers. The benefit: Each polynomial equation of degree n with real coefficients has exactly n complex roots. E.g. X^2 +1=0 has the two roots i and -i.

Whether complex numbers are understandable or not, depends on how familiar one is with complex numbers. From a mathematical point of view, complex numbers are necessary to solve problems from real numbers (solutions of polynomial equations) alike irrational numbers are necessary to solve geometric problem with rational numbers (diagonal of the unit square).

Irrational numbers are not irrational in the literal sense. Complex number are not complex in the literal sense. Imaginary numbers are not imaginary in the literal sense.

Added due to Frank's comment: The real-valued probability function is not sufficent because the fundamental equations of quantum mechanics and of all types of quantum field theories are wave-equations. A wave is characterized at each point in spacetime by its amplitude A and its phase phi, see John's answer. That property corresponds to the characteristic of a complex number z when written in polar coordinates:

+1 Good point in part 1 about counting. Parts 3-5 seemed off topic. In part 2, why is the real probability function not sufficient? I think that is what the OP wants to know.
– Frank HubenyMar 15 '18 at 12:04

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Regarding your comment that "Irrational numbers are not irrational in the literal sense"... On the contrary, irrational numbers are quite literally ir-RATIO-nal, assuming one uses the proper interpretation of what the literal meaning actually is. ;)
– David HMar 16 '18 at 7:49

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@David H Irrational numbers are non-rational numbers. A rational number is a fraction, i.e. the quotient of two integers.
– Jo WehlerMar 16 '18 at 7:57

"Complex number are not complex in the literal sense" -- sure they are: they have parts (a real part and an imaginary part) so they are literally complex rather than simple.
– John ColemanMar 16 '18 at 18:24

Are we answering the right question?

You touch upon an interesting point, but I have the feeling that your question isn't specific enough yet to reach a proper resolutions. Others have argued that 'complex numbers' aren't necessary for quantum mechanics. While I agree with their arguments, I think they're answering the question

and answer that, no, we can use some other mathematical object that isn't called that instead.

But that is a complicated answer to a trivial question, as I can simply define the 'lizard numbers' with exactly the same definition as the 'complex numbers' (without using that name, of course) and say you can simply describe QM using 'lizard numbers' instead. You might say that I'm cheating, but am I also cheating if my lizard numbers are different from complex numbers, but not very and can still be interchanged with the complex numbers to yield a valid theory of QM?

For example, suppose my lizard numbers extend the complex numbers with an l in addition to the i that indicates the 'lizardly axis' (in addition to the real and complex axis) but is usually set to 0 when performing QM, as there are no lizards on when working at a quantum scale (The lizardly axis is integral, as fractional lizards are animal cruelty).
Clearly, there are some issues that could be captured by asking better questions. An approach is this:

Is it possible to describe QM without using a mathematical structure that is 'essentially the same' as the complex numbers?

This question appears to represent the problem a bit better. However, it crucially depends on 1) what 'essentially the same' means and 2) what is a description of QM, or what is a physical description in general.

When are two mathematical objects 'essentially the same' for QM?

I think that you would agree that my lizard numbers yield an description of QM that is 'essentially the same', as I can simply replace every complex number by a lizard number and can keep the rest of the description. In the context of QM, it's not much more than a renaming, really.

But can we give a precise definition? If we are working within mathematics, I might come up with an approach. But we aren't in the realm of mathematics, but in physics and physics has some (mathematical!) problems that are 'widely regarded to be true' for which there is no mathematical proof (yet?). Take for instance the Yang-Mills gap hypothesis. The hypothesis has been confirmed to be sound by physical experiments and is part of standard theory, but this doesn't satisfy a mathematician (and perhaps some physicists), as this doesn't lead to a mathematical proof.

As we have seen that something can be proven in physics without proving it in mathematics, we really need a definition in physics. My knowledge on physics is lacking, so I'm unable to proceed here. But I doubt that an expert in physics would be able to give an unambiguous definition of what 'essentially the same' should mean here. (feel free to contradict me on that, though!)

When is something a 'description of QM'?

Contrary to the title, let's look at describing the quantum wave distribution, as that seems easier and is what the question is actually asking.
Still, this is perhaps even harder than the previous point. There exists descriptions of this function in different languages with different terms, so I suppose this should be 'independent of language', somehow. Also, do we take any lecture on this function to be a valid description? Probably not. We likely should require that the description allows us to unambiguously know how interpret the function in the results of physical experiments.

Can we conclude anything?

I hope that I have shown that the assertion that 'complex numbers are necessary to describe the quantum wave distribution function' is not as simple as it seems. Should we ask why something is true, before we know that it is true? Probably not, but then again, I know fairly little about philosophy. Perhaps these tricky questions have easy answers I'm simply ignorant of. If you know them, I'd be very glad to hear them, but this is all I can add.

If physical reality necessarily needs some absurd or undefinable mathematical construct at its fundamental level then we can say that theory can not be understood because at fundamental level theory can not be understood. Thanks for your answer. I have accepted the answer which resolved my query.
– Dheeraj VermaMar 15 '18 at 16:23

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It looks like there is a trivial isomorphism between the complex numbers and lizard numbers. They are the same up to isomorphism. How does this help answer the question?
– Frank HubenyMar 15 '18 at 16:27

@FrankHubeny No, there is a trivial isomorphism between lizard numbers with l=0. (at least, the second definition of lizard numbers) Just as there is a trivial isomorphism between quaternions (also suggested by the accepted answer) with the k and jdirection equal to zero and complex numbers. (which shows that the numbers aren't as crazy as I make them look.) Also, I'm not technically answering the question, but rephrasing it. I think that's not such a strange thing do in philosophy?
– Discrete lizardMar 15 '18 at 17:53

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@FrankHubeny Oh, wait I think you're referring to the lizard numbers 'of the first kind' The fact that this 'answers' the question trivially means that the question is suspect. The fact that you say they are 'the same' is precisely my point: we must be clear at what 'the same' means, but I also argue that some equivalence relation over mathematical structure is unlikely to be sufficient.
– Discrete lizardMar 15 '18 at 17:55

Physics does not define reality. Physics defines a model that approximates reality in a testable fashion. Reality can—and, going by experience, will—mandate we update or abandon any given model as we continue to test it. As such, the mathematics, such as complex numbers, are not part of reality in any provable way. They are part of the mathematical structures we use to construct the model. You are mistaking a toy car for a real car, loosely speaking.

More to the point, if you are assuming that physics, expressed using complex numbers among other things, literally defines reality, as your final question does, then the logical reason it uses something like complex numbers is "by assumption".

Moreover, no part of physics asserts that a complex number represents a measurable quantity. All physical operators have a real valued spectrum, and it is the spectrum of an operator which tells us the possible values we can measure. The complex numbers are background information that are solely a part of the particular mathematical model at hand. When you go to actually measure anything, you will only ever get real numbers. Your model that tries to explain why you measure the things you do may need more than that, but this is an artifice of your model and not of objective reality.

Complex numbers are ordered pairs of numbers that have an extended definition of multiplication that is useful for representing circular motion in two-dimensions. (The definition of multiplication for complex numbers represents rotation around the origin point, plus scaling of the amplitude of that point according to the normal rules of scalar multiplication.) So to say that complex numbers are "a part of reality" is, at best, just a short-hand way of saying that circular motion (and other similar wave-like motion) occurs commonly in reality, and so the mathematical tool that is tailored to describe this phenomenon tends to crop up a lot as a useful descriptive tool.

Remember that numbers (of any kind) are an abstraction that is used to describe concrete aspects of reality. To say that a mathematical object "is part of reality" is false in the concrete sense, but it can be true in the metaphorical sense that aspects of reality are accurately described by those abstractions. In the case of complex numbers, part of the confusion here comes from incorrect understanding of what they are ("but they're imaginary", etc.), which leads people to set them apart from other types of numbers, and imagine they their "existence" is somehow stranger than the "existence" of the real numbers, rational numbers, etc.

Imagine if you could only measure the heat produced in an a.c. circuit, and had no way to know the current. P=I^2R
You would only be able to get a positive quantity from an unobservable current that seemed to be able to 'unphysically' be positive & negative.

In this analogy power is just like any quantum observable, like position. And the 'unphysical' bit gives an underlying variable, but in this case one which cannot be observed, e.g. a spacial distribution of probabilities.

In an atom the observables are coupled together into an equation of state, the phase records angular momentum or spin. Spin can be up or down, in quantised amounts, but the spacial probability doesn't care which way it is facing, only the magnitude.

As I am not high enough level to comment, I will have to post an answer.

I think this comes down to the unfortunate use of calling part of the complex number imaginary and what this instills in a persons mind when first learning complex numbers.

But as others have tried to point out, people take for granted that the real number system is real - just because real is in its name and unquestioned, probably because of the age you are exposed to it compared to if you ever get exposed to Imaginary numbers or not.

"the quantum wave distribution function necessarily uses complex numbers to represent itself" - as others answered, this is not obvious, in the best case. However, others mostly argued that you can replace a complex number by two real numbers. On the other hand, one can use just one real wave function instead of the complex wave function, at least in some important general cases. The reason is modern physical theories are invariant under so called gauge transforms, so a complex wave function can generally made real by a gauge transform without changing the underlying physics. Schrödinger (Nature 169, 538 (1952)) showed that using the example of the Klein-Gordon equation in electromagnetic field (the Klein-Gordon equation is the simplest relativistic version of the famous Schrödinger equation). Schrödinger wrote: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." It turned out that the spinor wave function of the more realistic Dirac equation can also be replaced by one real function (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf - my article in the Journal of Mathematical Physics).

Nobody else seems to have addressed this point so here's something else to consider: of all the numbers you know, complex numbers are the only ones that form an algebraically closed field.

Consider natural numbers: if you want to solve the primary school problem of "how many apples does Alice get if Bob has 12 to start with and Charlie takes 5", you eventually realize that negative numbers are necessary. At first, negative numbers, as well as 0 as a number seem absurd to the untrained mind. But you quickly see that there's nothing weird or "unreal" about them... even though you'll never see "minus two apples" in real life.

Then you get into rational numbers, and quickly see that the "circle can't be squared", i.e. solving polynomials is not possible if you don't expand your group to irrationals as well. Not everything can be expressed as a quotient of two whole numbers. The seemingly innocuous a^2 + b^2 = c^2 equation, even though it is "obviously defined", doesn't work for a bunch of numbers a and b that are rational.

(This problem crops up in places like watch making where it's not always possible to create gears that exactly match the desired ratios - since gears can only have a natural number of teeth: you can only ever create rational ratios. This is why mechanical watches are said to be accurate to within x years: it doesn't indicate how well they keep time, rather how close the rational approximate is to the real number).

Point being: in all of these seemingly complete number sets, you can pose a problem that requires you to expand your definition of "what a number is" to something it didn't contain before in order to be able to solve it.

This is where complex numbers are special. Once you expand outwards and hit complex numbers, everything can be solved within that field. There exists no solution to any problem that requires you to use numbers outside of that field.

In that sense, Complex numbers are an integral part of reality because a right angle triangle exists regardless of what numbers you ascribe by, and similarly, the solution to a polynomial exists regardless of whether you believe in imaginary numbers or not. Complex numbers, as weird as they may be, actually solve all of our external math problems that deal with numbers.

As others have said, QM can be modeled using different numbers, but that's both true and irrelevant. The real insight is that on the totem pole of mathematical understanding, starting with basic counting skills you acquire as a child, you don't need to climb higher than complex numbers to solve all of your analytical needs.

Having said this, I'm sure a student of pure math will prove me wrong by informing me about an esoteric problem that requires a weird number field that I'd never heard of before.

Nice post. But I do not understand the apple problem. Who is giving Alice the apple? Why is the answer -2 (if it is)? Maybe I need a primary schooler to show me.
– Mark AndrewsMar 17 '18 at 23:51

The point isn't about how to teach pre-schoolers with apples, the point I'm making is about the solvability of various problems. To be able to solve 'basic' equations, you need negative numbers. To solve linear equations in a generalized manner, you need rational numbers. To be able to solve polynomials, your require irrationals. To solve the most complex of problems (e.g. differential equations) all you'll ever need is complex numbers. With regards to the apple question: "5 apples + how many apples equals 8 apples?" requires negative apples.
– user247243Mar 19 '18 at 3:02

If physics defines the physical reality, then what we are saying by the statement above is that the reality is made up of immeasurable and undefinable complex numbers.

This is an at least ~2400 years old argument going back to Plato, Aristotle et al: do mathematical objects (numbers etc.) exist physically, or are they just constructs in our mind?

A similar argument goes for language: does a word like "chair" exist, or does it not exist? I.e., does it have any physical meaning except for firing certain synapses in our head?

Another example: There are people who deny the existence of infinities like irrational numbers because they could not be constructed completely; they go to great lengths to build alternate mathematical buildings from scratch which do not need infinity.

Numbers can be understood. Every operation on the numbers can be understood. But complex number 'i' can not be understood. Initially the claim was , as I had learned in a you tube video , that without 'i' QM can not be constructed!! That was absurd as it made reality impossible to understand. Currently I am looking for real number based formalism of QM. I suspect it will lead to new insights as the issue has been resolved. It will increase our understanding of reality.
– Dheeraj VermaMar 16 '18 at 14:41

@DheerajVerma: i can surely be understood. It is a formalism like most if not all of mathematics. It is nothing mystical, magical, weird.
– AnoEMar 16 '18 at 15:20

Abstractly i is square root of -1. But tell me one instance of physical quantity or quality which is necessarily i and can not be represented using real numbers.
– Dheeraj VermaMar 16 '18 at 15:34

@DheerajVerma: if you don't like i you are free to change any calculation on whatever topic you are trying to solve to work without i. It is simply a tool, nothing more, nothing less, to make calculations easier. But so is _every_ other mathematical feature (like, e.g., infinitesimals, logarithms, integrals, etc.. You could to quantum physics without numbers at all, only by counting out long columns of "1"''s, doing everything while starting from first principles (axioms). It would be unfathomably hard, but whether you use i or not has nothing whatsoever to do with "reality".
– AnoEMar 16 '18 at 15:40

Yes I agree with u. That is exactly what I was expecting. Reality should always be theoretically understandable. Saying that wave function is necessarily unfathomable by declaring it to be necessarily a complex function was a wrong approach. Now the issue has been resolved. QM can be constructed using real numbers in principle.
– Dheeraj VermaMar 16 '18 at 15:46

Physics does not "describe reality". "Reality" is a metaphysical concept and is forever beyond experimental results. Physics gives relationships between observable situations. It relates one set of observations to another set of observations at a later time. It is OK for the wave function to be complex because it is not an observable quantity. (They are constructed out of statistical data and can be used to calculate statistical results but you can't observe one like we think of observing a baseball traveling as it moves, or sits.) The wave function is useful for relating one set of observations to another but it should not be considered as "describing reality" as such. In fact you can't assign physical properties to it in-between observations/measurements. That is the famous measurement problem. This really bothered John Bell who came up with a test for definite properties in-between observations/measurements. That hasn't gone well for assuming definite physical properties in-between observations.
I think that by definition there must be something corresponding to "reality" but it is nothing like what one might call "classical reality".

Thank you for your interest in this question.
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